Conversely, assume that (
x
𝑛
) is a sequence in
ℜ
𝑛
with
𝑥
𝑛
𝑖
→
𝑥
𝑖
for every coordinate
𝑖
.
Choose some
𝜖 >
0. For every
𝑖
, there exists some integer
𝑁
𝑖
such that
∣
𝑥
𝑛
𝑖
−
𝑥
𝑖
∣
< 𝜖
for every
𝑛
≥
𝑁
𝑖
Let
𝑁
= max
𝑖
{
𝑁
1
, 𝑁
2
, . . . , 𝑁
𝑛
}
. Then
∣
𝑥
𝑛
𝑖
−
𝑥
𝑖
∣
< 𝜖
for every
𝑛
≥
𝑁
and
∥
x
𝑛
−
x
∥
∞
= max
𝑖
∣
𝑥
𝑛
𝑖
−
𝑥
𝑖
∣
< 𝜖
for every
𝑛
≥
𝑁
That is,
x
𝑛
→
x
.
A similar proof can be given using the Euclidean norm
∥⋅∥
2
, but it is slightly more
complicated. This illustrates an instance where the sup norm is more tractable.
1.215
1. Let
𝑆
⊆
𝑋
be closed and bounded and let
x
𝑚
be a sequence in
𝑆
. Every
term
x
𝑚
has a representation
x
𝑚
=
𝑛
∑
𝑖
=1
𝛼
𝑚
𝑖
x
𝑖
Since
𝑆
is bounded, so is
x
𝑚
. That is, there exists
𝑘
such that
∥
x
𝑚
∥ ≤
𝑘
for all
𝑚
. Applying Lemma 1.1, there is a positive constant
𝑐
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Dr.DuMond
 Macroeconomics, Norm, Metric space, normed vector space, positive constant ????, subsequence ????????

Click to edit the document details